Multiple-constraint cooperative guidance based on two-stage sequential convex programming

Wei DONG, Qiuqiu WEN,*, Qunli XIA, Shengjing YANG

a School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

b Beijing Aerospace Technology Institute, Beijing 100074, China

KEYWORDS

Abstract An improved approach is presented in this paper to implement highly constrained cooperative guidance to attack a stationary target. The problem with time-varying Proportional Navigation (PN) gain is first formulated as a nonlinear optimal control problem, which is difficult to solve due to the existence of nonlinear kinematics and nonconvex constraints.After convexification treatments and discretization, the solution to the original problem can be approximately obtained by solving a sequence of Second-Order Cone Programming(SOCP)problems,which can be readily solved by state-of-the-art Interior-Point Methods (IPMs). To mitigate the sensibility of the algorithm on the user-provided initial profile, a Two-Stage Sequential Convex Programming (TSSCP)method is presented in detail.Furthermore,numerical simulations under different mission scenarios are conducted to show the superiority of the proposed method in solving the cooperative guidance problem.The research indicated that the TSSCP method is more tractable and reliable than the traditional methods and has great potential for real-time processing and on-board implementation.

1. Introduction

The rapid development of missile defense technology has driven the exploration of emerging cooperative guidance for multiple missiles. A typical cooperative guidance scenario, in which multiple missiles are required to simultaneously attack the same target, can effectively handle the threat of interceptors and greatly enhance the penetration capability of missiles.1–3Obviously, the impact time is a key constraint of a cooperative engagement. In addition, considering the limitations of the field-of-view of the seeker and the maneuverability of the missile, the impact angle, look angle and acceleration constraints must be taken into account.4,5Therefore,cooperative guidance is fundamentally an optimal control problem with multiple constraints, of which the optimization objective could be minimizing the required magnitude of the lateral acceleration during flight.6

To address this optimal control problem, various approaches have been suggested over the past few years;these approaches can be classified into two major categories:analytical methods and numerical methods. The former is dedicated to finding a closed-form solution, namely, the guidance law,based on modern control theory, while the latter focuses on solving the problem by the effectiveness of mature optimization algorithms.The time-constrained guidance law was introduced for the first time by Jeon et al.7,where an Impact-Time-Control Guidance (ITCG) law combining the Proportional Navigation(PN)law and the feedback of the impact time error was proposed based on the maximum principle.Lee et al.8further extended the ITCG law to the Impact Time and Angle-Control Guidance (ITACG) law, which realized the concurrent control of the impact time and impact angle. A novel Time-Constrained Guidance (TCG) law was designed by applying a virtual leader scheme and stability method.9The core idea of the virtual leader scheme was to adopt a virtual leader for real missiles to convert a guidance problem with time constraints to a nonlinear tracking problem,thereby making it possible to settle the problem with a variety of control methods.To satisfy the impact constraints, a line-of-sight rate shaping process was introduced to the sliding mode-based ITACG law.10Zhang et al.11proposed a closed-form guidance law with impact time and angle constraints, where feedback control was added to the Biased Proportional Navigation Guidance (BPNG), and then the actual time-to-go tracked the designated time-to-go.

However, for the abovementioned analytical methods, the ignorance of the look angle and acceleration constraints reduces the attack precision and performance and therefore limits their application.Due to the existence of multiple rigorous constraints, it is difficult to address the cooperative guidance problem in an analytical manner. In recent years,approaches based on computational methods have gained much attention for solving guidance problems, such as the pseudospectral method12,13, nonlinear programming14and receding time domain optimization.15,16Specifically, Fang et al.17translated the cooperative guidance problem into a nonlinear optimal control problem with a fixed terminal horizon and terminal state constraints by constructing mathematical models, and subsequently solved it via Sequential Quadratic Programming (SQP). Zhao et al.18presented a time-constrained guidance approach for a multimissile network using the nonlinear Model Predictive Control (MPC)technique.Li19transformed the cooperative guidance problem under the constraints of the maneuverability limit and minimum collision avoidance distance into a nonlinear optimal problem with fixed-time terminal state constraints and solved it by adopting the Gaussian pseudospectral method (GPM).However,the high computational cost impedes the online optimization of these approaches.

In recent years, benefiting from the great potential of realtime processing and the capability to handle various constraints in optimal control problems, convex optimization has gained increasing popularity in addressing aerospace guidance and control problems.20–23As a subclass of convex optimization, Second-Order Cone Programming (SOCP), where the cost function is linear and is subject to linear equality and second-order cone constraints24, could be efficiently and reliably solved by the Interior-Point Methods(IPMs).25Nevertheless,the vast majority of aerospace engineering problems do not naturally possess the specific form required in SOCP.Reasonable convexification techniques are required to first convert them into a framework that can be discretized as SOCP problems. Thereafter, the approximate solutions of the original problems can be obtained based on the Sequential Convex Programming (SCP) method.26However, the cooperative guidance problem studied in this paper is subject to multiple constraints, including the impact time, impact angle, look angle and lateral acceleration.In the treatment of convexification, the over conservative approximation of nonconvex constraints may lead to an infeasible problem. Herein, an improved approach referred to as Two-Stage Sequential Convex Programming (TSSCP) is proposed to mitigate the sensibility of the algorithm on the initial profile and convexification techniques to promote the computational efficiency and tractability.

The paper is organized as follows. Section 2 presents the preliminaries of the guidance geometry and optimal control problem with a time-varying PN gain. In Section 3, the treatments to convexify the original problem are investigated, followed by a detailed description of the proposed TSSCP method in Section 4.The effectiveness of the method is demonstrated by numerical simulations in Section 5. Finally, concluding remarks are presented in Section 6.

2. Preliminaries

2.1. Guidance geometry

Cooperative guidance for multiple missiles can be categorized as either‘‘open-loop”or‘‘closed-loop”.27The former needs to set a common impact time at the beginning of the terminal guidance, and thereafter each missile independently attacks the target subject to the impact time.In the latter case,all missiles attempt to realize a salvo attack via transmitted and shared information during the engagement; hence, the impact time does not need to be preset.Communication between missiles plays a key role in closed-loop cooperative guidance.However, considering actual combat scenes, the adversary can disrupt communication by resorting to ElectroMagnetic Interference (EMI) technology. Thus, in terms of reliability,open-loop cooperative guidance possesses more theoretical necessity and practical significance.

Fig.1 Homing guidance geometry.

In open-loop cooperative guidance,due to the relative independence of each missile, the cooperative guidance problem is essentially a guidance problem constrained by the impact time,as well as other constraints, such as impact angle, look angle,and acceleration.The guidance for missileito attack a stationary target on the horizontal plane is separately investigated with the homing guidance geometry shown in Fig.1, whereriand λiare respectively the Line-of-Sight (LOS) range and LOS angle between them,Viis the missile velocity,aiis the lateral acceleration, γiis the flight-path angle, and εiis the look angle.

To facilitate the engagement kinematics, the following assumptions are made in this paper, as commonly done in other relevant literature:

(1) Both the missile and the target are considered geometric points in the planar plane.

(2) The seeker and autopilot dynamics of each missile are much faster in comparison with the guidance loop so that their lags are negligible.

(3) The velocity magnitude of each missile is constant throughout the engagement span, and the lateral acceleration is perpendicular to the velocity vector.

The kinematic equations of motion can be written as

where d(*)/dtrepresents the derivative of*with respect to timet.

In the terminal guidance problem, the Proportional Navigation (PN) guidance law is a common practice, because it not only provides guidance command information but also reflects the relationship between the flight-path angle rate and the LOS angle rate. The PN law calls for the commanded flight-path-angle rate to be proportional to the LOS angle rate,that is,

where the gainNPN,i(t) is a time-varying variable to be optimized, which is distinct from the traditional PN law, where the gain is a constant.

Hence, the kinematic equations of motion with the PN law can be described as

The lateral acceleration of missileican be computed by

Note that the LOS rangerishould decrease to zero at the terminal timetfto intercept the target, and it can be directly measured by the radar seeker during flight. If we choose the LOS rangerias the independent variable, the conversion and analysis of the optimization problem will be simpler in later sections. According to this idea, Eq. (3) can be rewritten as

At this point, the differentiations are now with respect to the LOS rangeri.In the remainder of this paper,for simplicity of notation,the subscriptiand argumentriwill be suppressed,for example, the range-varying variable εi(r) →εi(ri) is abbreviated as ε.

2.2. Optimal control model

The PN gainNPN, as already mentioned, is a variable to be optimized.To find the most appropriate PN gain that not only satisfies all constraints but also offers the optimal performance of the guided trajectory,the optimal control model of multipleconstraint guidance is established in this subsection.

In addition to the engagement kinematics Eq. (5), the constraints that must be satisfied in the guidance problem can be divided into two broad categories:Process constraints and terminal constraints.

2.2.1. Process constraints

Process constraints refer to constraints that cannot be violated during the flight, depending on the inherent structure and characteristics of the missile. According to the problem analyzed in this paper, process constraints include the lateral acceleration constraint and look angle constraint. The former stems from the maneuverability limitation of the missile, and the latter maintains a seeker lock-on to a target. These constraints can be expressed as

whereamaxand εmaxindicate the maximum allowable magnitudes of the lateral acceleration and the look angle, respectively, and εmax＜π/2 is assumed in this paper to ensure that the LOS rangermonotonically decreases during the engagement.

2.2.2. Terminal constraints

To simultaneously impact the target with multiple missiles, an impact time constraint is required, that is,

At the same time, each missile is required to intercept the target from a specified direction for maximum warhead effectiveness or lethality, that is,

Furthermore, to guarantee a terminal zero miss distance and ensure that the terminal velocity directly points to the target, the following terminal constraints must be satisfied.

Under the basic premise that the aforementioned constraints are satisfied, a reduction in the lateral acceleration magnitude is expected. Thus, the desired performance index to be minimized is chosen as

This equation is converted into a form with the LOS rangeras the independent variable

Thus far, with the nonlinear kinematics in Eq. (5) and the constraints in Eqs.(6)–(11),we have a highly constrained nonlinear optimal control problem Π0.

Notably, problem Π0is inherently a NonLinear Programming (NLP) problem. Directly solving this problem using NLP algorithms will lead to an unknown computational time,a lack of guaranteed algorithm convergence, and even infeasibility.28However, for cooperative guidance, the optimization problem must be solved in real time so that it can be utilized in on-board applications.29The underlying idea is using an SOCP-based method to solve this problem, which can meet the enormous demands for reliability and celerity. The challenge of this endeavor lies in how to formulate the subproblem so that it can be readily solved by state-of-the-art IPMs without loss of performance or solution fidelity. The next section will elaborate on the treatment to transform the original problem Π0into a second-order cone programming problem.

3. Convexification treatments

In this paper,the solution to nonconvex problem Π0with nonconvex inequality or equality constraints is approached by a successive solution process in which a sequence of SOCP problems are solved. To convert the original problem Π0into a framework that can be discretized as a SOCP problem, all of the nonconvex constraints must be transformed into convex ones by reasonable convexification techniques. Notably, the following convexification treatments are based on the assumption that the last successive solution before the current iteration has already been obtained.

3.1. Linearized kinematics

As mentioned above, The SOCP problem requires that its equality constraints be linear.Thus,to become linear algebraic equations after discretization,the kinematic equations must be transformed into linear ones. If we define a new state variable σ=tan ε and letx=[λ,σ,t]be the state vector andu=NPNbe the control variable,then the kinematics in Eq.(5)will take the form

where

where

InA(x(k)),

An additional trust-region constraint must be satisfied as follows

where δ is a constant column vector specified manually.Notably, the preceding inequality equation is a componentwise equation for each state variable. The introduction of the trust-region constraint maintains the validity of the preceding linearization; otherwise, the linearization of the kinematics is invalid.

3.2. Convexification of the performance index

Substituting σ=tan ε into Eq. (13) gives

Minimizing the above performance index is equivalent to minimizing

subject to

where η is the slack variable. Thus far, the performance index Eq. (22) has a linear integrand. The inequality constraint Eq.(23) will be convexified in the following subsection.

3.3. Convexification of the inequality constraints

By bringing in the new state variable σ,the process constraints Eqs. (6) and (7) can be converted to

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Eq.(25)is a second-order cone constraint,which is convex.The constraint expressions Eq.(23)and(24)are highly nonlinear and should be further treated by the successive approximation technique.21Defining

Then, Eqs. (23) and (24) are transformed as follows

After the preceding convexification treatments,the solution of problem Π0can be obtained by solving the following convex optimal control problem Π1in sequence.

4. Two-stage sequential convex programming

4.1. SOCP subproblem establishment

Inspired by the idea of the point collocation method, the continuous LOS range in problem Π1is discretized uniformly intoNintervals in [r0;rf] with a step size of Δr=(rf-r0)/N.The LOS ranges at discretized points are denoted as｛r0，r1，...，rN｝ and are compatible with the condition ofri=r0+iΔr,i=0, 1, ...,N. The continuous statexand controluof problem Π1are discretized intoxi=x(ri) andui=u(ri), and all constraints are enforced at these discretized points.

To convert it into a linear equality constraint at discretized points, the differential equation Eq. (16) is numerically integrated by the Euler formula as follows:

The state vector,control variable and slack variable at each discretized point can be denoted together as an optimization variable vectorz, that is,

After the preceding discretization treatment, problem Π1is converted into the SOCP problem ΠSOCP.ΠSOCP: Minimize Eq.（32） Subject to Eq.（30）

4.2. Proposed TSSCP method

The solution of the original problem Π0is approximately obtained by solving a sequence of SOCP problems ΠSOCPfork=0,1,2,...until the solution converges;this procedure is known as SCP.In the process of the SCP method,the SOCP problem to be solved at the current iteration,except at the first iteration, is established based on the solution obtained at the previous iteration. Suppose the initial state guess is chosen asx(0). The SCP method will then first solve the problem established byx(0).However,because the original problem is highly constrained and the treatment of nonconvex constraints is approximate, the problem at the first iteration may be infeasible, which leads to the termination of the procedure. Furthermore, Liu et al. deemed that once the convex and concave constraints are satisfied in an iteration of the SCP, they will be satisfied in later iterations under mild conditions.26That is,if the initial state guess does not satisfy the constraints,then the method to convexify the constraints and the choice of the initial guess can affect the robustness of the convergence of the successive solution procedure.21

To mitigate the sensibility of the algorithm on the initial state profile31, we propose a TSSCP method, of which the flowchart is given in Fig.2.

(1) Setk=0, and assign an initial state profilex(0).

(3) Check whether the solution satisfies the following convergence condition

where ζ is a sufficiently small constant vector defined manually, which refers to the maximum convergence error. If the above condition is met, then go to step (4), otherwise setk=k+1 and go back to step (2).

(4) The solution to the SOCP problem is｛x（k+1）;u（k+1）;η（k+1）｝.

Fig.2 Flowchart of the proposed TSSCP method.

Note that for the proposed TSSCP method,we need to provide only an initial state profilex(0),rather than an initial guess of all optimization variables including μ(0)and η(0),because the initial state profile is used to compute only thex(k)-dependent parameters at the first iteration.Therefore,the initial state profile can be assigned in a relatively free and flexible way without satisfying all conditions,for instance,simply setting the profile as a straight line from the initial value to the terminal value.In addition, the desired impact timetfand impact angle λfmust be reasonable values within the maneuverability of the missile,or otherwise there will be no feasible solution no matter what method is applied. The proposed TSSCP method is a way to avoid infeasibility by the SCP method but also preserve its high efficiency. The following extensive numerical tests will instill confidence in the superiority of the proposed method.

Remark 1.Insomespecialorsimplifiedcases,theconvergence ofsuccessivesolutionprocedurescanbeproventheoretically.For example,BanksandDinesh.32provedtheconvergenceofthe solutionsequenceforaclassofoptimalcontrolproblemswitha quadraticperformanceindexandwithoutanyconstraints.However,forthevastmajorityofoptimalcontrolproblems,duetotheirnonlinearityandcomplexconstraints,astrict theoreticalproofoftheconvergenceofthesuccessivesolution sequenceismuchmoredifficulttoconduct.33Toprovethe convergence,theexistingstudiesmostlyresorttonumerical experiencesinsteadoftheoreticalproofs.34Inthenextsection,somenumericalsimulationswillbepresentedtoshowthe convergenceperformanceoftheproposedmethod.

5. Numerical simulations

In the following simulations,the discretized interval numberNis set as 100,and the initial state profiles are all simply set to be straight lines from the initial values to the terminal values.The parameters of the trust-region constraint in Eq. (35) and the convergence condition in Eq. (41) are set as

δ= [60π/180， tan（6 0π/180）， 10]T

ζ= [0.1π/180， tan（0 .1π/180）， 0.01]T

MATLAB R2016b software with the YALMIP35and MOSEK36toolboxes is used to program and solve the SOCP problems. All the results are obtained on a personal computer with an Intel Core i7-6700HQ CPU @ 2.60 GHz and 4 GB RAM.

Table 1 Objects and mission scenarios of three cases.

Without loss of generality, three cases containing different objects and mission scenarios are conducted in this section,as shown in Table 1.

5.1. Case 1: algorithm validity verification

To verify the effectiveness and convergence of the proposed algorithm, we consider an air-to-ground missile to vertically impact a stationary target at a specified impact time. Suppose the initial LOS ranger0is 8 km, the initial LOS angle λ0is-10°, and the initial look angle ε0is 10°. The velocityVof the missile remains constant at 280 m/s during flight. The expected impact angle λfand the impact timetfare 90° and 35 s, respectively. The constraint parameters are set as εmax=45° andamax=5g.

This problem is infeasible if directly solved by the SCP method but is feasible using the TSSCP method,which proves that the proposed method is more reliable with respect to SCP.Specifically, the converged solution takes a total of 15 successive iterations to be obtained, in which the first stage costs 8 iterations and the second stage costs 7 iterations. It takes the IPMs 0.332 s to solve the whole problem with a short time ranging from 0.015 s to 0.025 s to solve the SOCP subproblem in each successive iteration. The profiles of the variables in all successive iterations are demonstrated in Figs. 3 and 4. The solutions converge gradually with the increasing iterations.After 4 iterations for each stage, the profile deviation between two adjacent iterations is already almost indiscernible to the scale of the figure. From Figs. 3(d), 4(b), (d) and (f), the constraints including the terminal constraints of the impact time and angle, as well as the process constraints of the look angle and lateral acceleration, are well satisfied after convergence.

To verify the proposed TSSCP method, the GPM using GPOPS-II37software is employed to solve the same problem,which takes 6.568 s. The contrast profiles of the solution obtained by the TSSCP method and GPM are shown in Fig.5. The results indicate that the two methods yield almost the same results,where the relatively small differences could be attributed to the different discretization schemes and model treatment techniques used in the two solution approaches.Thus,the proposed TSSCP method provides comparable optimality to the GPM and possesses a higher efficiency for cooperative guidance.

Remark 2.ItcanbeseeninFig.3(b)thattheNPNtakes negativevaluesatthebeginningoftheflightandthenremains positiveforanincrediblylongtimeinthefollowingflight,which isquitedifferentfromtheconventionalPNguidancelawin whichNPNisaconstantintherangefrom2to5.Physically,this situationoccursbecausetheinitialnegativeNPNcanincreasethe flighttrajectoryandtimetosatisfytheimpacttimeandangle constraints,whichisconsistentwiththeCooperativeProportionalNavigation(CPN)proposedbyJeonetal.38,wherethe navigationgainofamissileisnegativeatthebeginningtorealize salvoattack.

5.2. Case 2: salvo attack for multiple missiles

Salvo attack, as already mentioned at the beginning of this paper, achieves better performance in detecting targets, penetrating defense systems, and surviving threats.This subsection aims to demonstrate the effectiveness of the proposed TSSCP method under the salvo attack scenario. Three groundlaunched missiles are considered in this subsection, whose initial conditions,velocity magnitudes and constraint parameters are shown in Table 2. The simulation results are presented in Figs. 6 and 7.

Fig.3 Profiles of PN gain and acceleration in all successive iterations of proposed method.

Fig.4 State variables profiles in all successive iterations of proposed method.

The desired impact angle and impact time are achieved for each missile. Obviously, a missile with a smaller initial LOS range has a higher flight altitude to increase its flight-path length, as illustrated in Fig.6(a). The terminal constraints of the LOS angle and look angle, as well as the process constraints of the look angle and lateral acceleration,are satisfied,as shown in Figs. 7(a)–(c). Combining Figs. 6(b) and 7(d), it can be concluded that the PN gain profiles greatly differ for the various initial LOS ranges, especially at the beginning.Furthermore, a smaller initial LOS range renders the PN gain smaller at the beginning, which results in a rapidly narrowing gap between the LOS ranges of the three missiles at the beginning. Then, with a decrease in the LOS range, the optimized PN gains quickly converge and become consistent, which ensures excellent agreement regarding the impact time.In summary,the three missiles can achieve salvo attack under the proposed TSSCP-based cooperative guidance.

5.3. Case 3: successive attack for multiple missiles

The extensive application of precision-guided weapons in warfare has greatly stimulated the development of underground defense facilities. Targets with important strategic values have been transferred underground, and defense structures have become increasingly sturdy.Therefore,in the mission to strike hard targets buried underground, a greater penetration attack capability is required. For an individual missile, achieving as vertical an attack as possible is an effective means to enhance its penetration attack capability. However, for multiple missiles, in addition to a large-impact-angle attack for each missile, a successive attack with a certain time interval is expected. To confirm the application in successive attacks of the proposed TSSCP method, four air-to-ground missiles are considered in this subsection, whose parameters are shown in Table 3. Figs. 8 and 9 give the results of the simulation.

From Fig.8, four missiles successfully achieve a successive attack every 5 s using the proposed TSSCP-based cooperative guidance. Fig.9 demonstrates that all terminal and process constraints are satisfied for each missile. A larger expected impact time leads to a larger look angle maximum value, a smaller PN gain at the beginning and a longer flight path.By further analysis, the different PN gains at the beginning facilitate a change in the LOS range from the same initial value to gradually holding the fixed disparities. The existence of the LOS range disparities guarantees successive attacks, which is different from the salvo attacks in Case 2. The proposed TSSCP method can also be applied to implement successive attacks for multiple missiles.

Fig.5 Comparison of the solutions obtained by proposed method and GPM.

Table 2 Parameter settings of salvo attack.

Fig.6 Trajectories and PN gain profiles of salvo attack obtained by proposed method.

5.4. Computational time comparison

To further reveal the efficiency of the presented method, the computational times of the above three cases solved by TSSCP and GPM are recorded in Table 4.

From Table 4, the proposed TSSCP approach shows close to real-time performance,i.e.,below 0.5 s,even in a MATLAB implementation. It is believed that a greater computational speed can be achieved with a C/C++ implementation. In addition, for an online application of the proposed method,the initial state profilex(0)in the current optimized cycle could be assigned as the converged solution in the previous optimized cycle, rather than a straight line, which contributes to further reducing the computational time.Accordingly,the proposed TSSCP method has enormous superiority, especially in terms of computational time,in solving the multiple-constraint cooperative guidance problem.

Fig.7 Other variable profiles of salvo attack obtained by proposed method.

Table 3 Parameter settings of successive attack.

Fig.8 Trajectories and PN gain profiles of successive attack obtained by proposed method.

Remark 3.Forconvenienceofmodeltreatmentandspeedof optimization,themodelandsimulationarebasedonthe assumptionthatthemissilevelocityisaconstant,whichis commonlyconsideredintheresearchontheguidanceproblem.39First,forthemajorityofmissiles,thevelocityintheterminal guidancephasetypicallyhasarelativelysmallmagnitudetobegetalongguidancetime,satisfyingtherequirementofazero miss-distance.Thedragforceduringthisphaseissmall,andthus thevelocityvarieslittle.Additionally,whenappliedinpractice,theproposedTSSCPmethodcanbeusedinconjunctionwith ModelPredictiveControl(MPC)toeliminatetheeffectofthe time-varyingvelocity.40Ineachguidancecycle,theinstantaneousvelocityatcurrentmomentistakenintothemultipleconstraintcooperativeguidanceproblem,andthentheTSSCP methodisusedtoobtaintheoptimalPNgain,whichistakenasthecommandedPNgaininthenextguidancecycle.Anexcessive computationaltimehinderstheapplicationofthenonlinear programmingmethod,whereastheproposedTSSCPmethod makesonlineimplementationpossiblebecauseofitsaccuracy andeffectiveness.

Fig.9 Other variable profiles of successive attack obtained by proposed method.

Table 4 Computational times of the three cases.

6. Conclusions

In this paper, the cooperative guidance problem to attack a stationary target,constrained by multiple constraints including the impact time,impact angle,look angle and lateral acceleration,is formulated as an optimal control problem,which is difficult to solve due to the existence of nonlinear kinematics and nonconvex constraints. After convexification treatments and discretization,the original problem is converted into an SOCP problem,which can be readily solved by state-of-the-art IPMs.Eventually, to mitigate the sensibility of the algorithm on the initial profile, an improved approach called TSSCP is presented. Analyses of the numerical simulations provide the following conclusions.

(1) Compared with the existing time-constrained guidance laws, the proposed TSSCP method contributes to providing a more rigorous and systematic cooperative guidance scheme, which ensures the satisfaction of all constraints.

(2) Compared with the SCP method, the proposed TSSCP method is more tractable and reliable in that it is much more insensitive to the initial state profile, rendering it effective in highly constrained but feasible scenarios such as in salvo and successive attacks.

(3) Compared with nonlinear programming methods, such as the GPM, the proposed TSSCP method possesses much higher efficiency; for example, it solved the cooperative guidance problem within 0.5 s, proving its great potential for real-time processing and on-board implementation.

Acknowledgment

This study was supported by the Joint Foundation of the Ministry of Education of China (No. 6141A02022340).